Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

• Published in: Mathematics (Basel) Vol. 11; no. 17; p. 3714
• Main Authors: Liu, Kemeng, Li, Zewen, Pei, Donghe
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-09-2023
• Subjects: Classification; Euclidean geometry; Euclidean space; Food science; framed surfaces; Geometry, Plane; Geometry, Solid; Mathematical research; singularities; surfaces of osculating circles; Topological spaces; United States
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11173714

In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps.